I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows:
Let H be a Hilbert space. We are to prove, in two distinct ways, that if $ T \in B(H) $ is a contraction (Contraction: An operator T satisfying $ ||T|| \leq 1 $) which is not unitary, and if V is an isometric dilation of T, then the larger Hilbert space K, satisfying $ H \subset K $, is necessarily infinite dimensional.
Clarification: By isometric dilation I simply mean that the extended operator is itself an isometry.
Edit: the context for this is the Sz.-Nagy's dilation theorem as seen here https://en.wikipedia.org/wiki/Sz.-Nagy%27s_dilation_theorem And also from the text book by Nagy: Here is an arxiv link to this result on page 1 the Nagy theorem: http://arxiv.org/pdf/1012.4514.pdf
I unfortunately have to say that I truly have no idea on this one even after working hard at this, I don't have one way to prove this claim even in one method I thought assuming to get contradiction that I can assume K is finite dimensional and then work with matrices but I got nothing. I am stuck and truly desperate.